Field-induced reorientation of colloidal particles is especially relevant to manipulate the optical properties of a nanomaterial for target applications. We have recently shown that surprisingly feeble external stimuli are able to transform uniaxial nematic liquid crystals (LCs) of cuboidal particles into biaxial nematic LCs. In the light of these results, here we apply an external field that forces the reorientation of colloidal cuboids in nematic LCs and sparks a uniaxial-to-biaxial texture switching. By Dynamic Monte Carlo simulation, we investigate the unsteady-state reorientation dynamics at the particle scale when the field is applied (uniaxial-to-biaxial switching) and then removed (biaxial-to-uniaxial switching). We detect a strong correlation between the response time, being the time taken for the system to reorient, and particle anisotropy, which spans from rod-like to plate-like geometries. Interestingly, self-dual shaped cuboids, theoretically considered as the most suitable to promote phase biaxiality for being exactly in between prolate and oblate particles, exhibit surprisingly slow response times, especially if compared to prolate cuboids.